# Properties

 Label 2-336-7.4-c5-0-35 Degree $2$ Conductor $336$ Sign $-0.968 + 0.250i$ Analytic cond. $53.8889$ Root an. cond. $7.34091$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (4.5 − 7.79i)3-s + (−11.0 − 19.1i)5-s + (−126. − 26.6i)7-s + (−40.5 − 70.1i)9-s + (208. − 360. i)11-s + 797.·13-s − 198.·15-s + (687. − 1.19e3i)17-s + (1.15e3 + 2.00e3i)19-s + (−778. + 869. i)21-s + (−477. − 827. i)23-s + (1.31e3 − 2.28e3i)25-s − 729·27-s − 7.03e3·29-s + (630. − 1.09e3i)31-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s + (−0.197 − 0.341i)5-s + (−0.978 − 0.205i)7-s + (−0.166 − 0.288i)9-s + (0.519 − 0.899i)11-s + 1.30·13-s − 0.227·15-s + (0.577 − 0.999i)17-s + (0.734 + 1.27i)19-s + (−0.385 + 0.430i)21-s + (−0.188 − 0.326i)23-s + (0.422 − 0.731i)25-s − 0.192·27-s − 1.55·29-s + (0.117 − 0.204i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.968 + 0.250i$ Analytic conductor: $$53.8889$$ Root analytic conductor: $$7.34091$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{336} (193, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :5/2),\ -0.968 + 0.250i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.316998916$$ $$L(\frac12)$$ $$\approx$$ $$1.316998916$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-4.5 + 7.79i)T$$
7 $$1 + (126. + 26.6i)T$$
good5 $$1 + (11.0 + 19.1i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (-208. + 360. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 - 797.T + 3.71e5T^{2}$$
17 $$1 + (-687. + 1.19e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (-1.15e3 - 2.00e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (477. + 827. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + 7.03e3T + 2.05e7T^{2}$$
31 $$1 + (-630. + 1.09e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (4.88e3 + 8.46e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 5.40e3T + 1.15e8T^{2}$$
43 $$1 + 1.96e4T + 1.47e8T^{2}$$
47 $$1 + (-1.02e3 - 1.78e3i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (9.01e3 - 1.56e4i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-3.71e3 + 6.43e3i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (1.74e3 + 3.02e3i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (-7.92e3 + 1.37e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 5.81e4T + 1.80e9T^{2}$$
73 $$1 + (1.95e4 - 3.38e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-4.88e3 - 8.45e3i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 - 7.03e4T + 3.93e9T^{2}$$
89 $$1 + (7.21e4 + 1.24e5i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 + 7.93e4T + 8.58e9T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.21634098439687661354715587828, −9.226383918285151176647271872339, −8.455958252020127116657901535127, −7.46045476203689725062209569663, −6.37797623537009223601277914757, −5.60952194428769375511132234178, −3.83235626691317496463558393072, −3.16981203884020678710840393660, −1.40741748889352541167801562551, −0.34035693491137297772179044406, 1.54572592599928544773981665960, 3.18652749002496854757248991718, 3.79375999447260143362188845650, 5.22573297389418479673743106928, 6.39871181117403138415471448182, 7.25176614841748258323871493723, 8.530678442921327649579601560416, 9.362013074810042908633863679676, 10.10197503684529526236258207510, 11.08292648495588681860366588438